A vibration source is commonly coupled to a receiving structure by a vibration isolator. A key trade-off in the choice of vibration isolator is the requirement for a wide frequency range of isolation without excessive static deflection. This compromise can, in principle, be circumvented by employing a softening nonlinear isolator that presents a high stiffness to the weight of the isolated mass but a low tangent stiffness in the vicinity of the equilibrium position. The first part of this thesis is concerned with the static response of a number of elements that are expected to exhibit such a nonlinear stiffness characteristic. A mechanism with geometrical nonlinearity is studied first and found to offer some benefits compared with a similar one reported in the literature. Beams are commonly employed as linear springs and their suitability as nonlinear isolators is considered here. It is shown that the stiffness of a simply supported beam loaded transversely at its centre is of a hardening type in contrast to what is reported in the literature. Post-buckled beams are also investigated as candidates for nonlinear springs of vibration isolators although the sudden change in stiffness at the buckling point is unfavourable. Curved beams and beams with eccentric loading are investigated as alternatives to a straight axially loaded post-buckled beam. Static analyses are presented which show that curvature or eccentricity in loading can be incorporated to smooth the force-deflection curves. A commercially available rubber isolation mount is also studied as an example of an axially loaded curved element and its force-deflection characteristic measured. The hypothesis that it can be modelled by a curved beam is found not to hold. The inter-variability observed between samples is evaluated which illustrates the potential for mistune of nonlinear mounts in general. Nonlinear stiffness gives rise to the possibility of asymmetry about the equilibrium position, either as an inherent characteristic of the isolator or as a result of a mistuned added mass or static preload. A nonlinear isolator with asymmetric stiffness is modelled as a Duffing oscillator. The force transmissibility of the oscillator is obtained analytically using the Harmonic Balance Method from which the performance of the isolator is evaluated quantitatively as a function of both static load and mistuned mass. A study is presented for the case of a nonlinear isolator comprised of a curved beam. The high stiffness of the beam in extension causes impulsive behaviour in the transmitted force which is alleviated by the inclusion of a linear spring placed in series. It is shown that this combination significantly outperforms a linear isolator with the same static deflection.